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Single boson exchange decomposition and approximation

The single boson exchange (SBE) decomposition is an alternative way to decompose the full vertex FF into diagrammatic contributions, based on the idea that many interaction processes can be effectively described by the exchange of a single bosonic particle (e.g., a spin or charge fluctuation).

Interaction reducibility

This framework employs the concept of interaction reducibility (also called UU-reducibility), which is related but distinct from the two-particle reducibility used in parquet theory. Any given diagram is called UU-reducible if it can be separated into two parts by removing a single bare internal interaction vertex F0F_0. There are three distinct channels in which this can happen, corresponding to the same channels as in two-particle reducibility: the particle-hole (phph), the transverse particle-hole (ph‾\overline{ph}), and the particle-particle (pppp) channel. Indeed, any diagram that is UU-reducible in channel rr is also two-particle reducible in that channel. The inverse is not true, however.

SBE decomposition

The sum of all diagrams that are UU-reducible in channel rr is denoted by Δr\Delta^r, while the sum of all diagrams that are not UU-reducible in any of the three channels is denoted by ΛU\Lambda^{U}. The full vertex FF can then be decomposed as

F=ΛU+∑r∈{ph‾,pp,ph}Δr−2F0 .\begin{align} F = \Lambda^{U} + \sum_{r \in \{\overline{ph}, pp, ph\}} \Delta^{r} - 2F_0 \, . \end{align}
(1)

The bare interaction vertex F0F_0 is formally UU-reducible in all three channels and therefore contained in each of the three Δr\Delta^r. To avoid double counting, we therefore need to subtract 2F02F_0.

Similarly to the parquet theory, one can also define the UU-irreducible vertex in channel rr as

Tr=F−Δr=ΛU+∑r′≠rΔr′−2F0 .\begin{align} T^r = F - \Delta^r = \Lambda^{U} + \sum_{r' \neq r} \Delta^{r'} - 2F_0 \, . \end{align}
(2)

SBE equations

The main strength of the SBE decomposition over the parquet decomposition lies in significant simplifications in the case where the bare interaction F0F_0 is local (i.e., momentum-independent) and instantaneous (i.e., frequency-independent), (as, e.g., in the Hubbard model). As in the chapter on two-particle channels, it is in this case useful to introduce a unit vertex for non-frequency-momentum quantum numbers in channel rr, 1r\mathbf{1}^r, which satisfies F=1r∙F=F∙1rF = \mathbf{1}^r \bullet F = F \bullet \mathbf{1}^r, where the symbol ∙\bullet denotes contractions over all quantum numbers, indices and other degrees of freedom except frequency (and momenta).

One then defines the Hedin vertices γr\gamma^r as

γr=1r+1r∘χ0r∘Trγ‾r=1r+Tr∘χ0r∘1r ,\begin{align} \gamma^r &= \mathbf{1}^r + \mathbf{1}^r \circ \chi^r_0 \circ T^r \\ \overline{\gamma}^r &= \mathbf{1}^r + T^r \circ \chi^r_0 \circ \mathbf{1}^r \, , \end{align}
(3)

where χ0r\chi^r_0 is the dressed bubble in channel rr. The Hedin vertices are UU-reducible in channel rr and, crucially, depend only on two frequencies (or momenta), since the dependence on the third frequency is trivially contracted away through the ∘ contraction with 1r\mathbf{1}^r.

The Hedin vertices describe the effective coupling between fermionic particles and collective bosonic fluctuations in channel rr, as can be seen from the main result of SBE theory: The UU-reducible diagrams in channel rr can be written as

Δr=γ‾r∙Wr∙γr ,\begin{align} \Delta^r = \overline{\gamma}^r \bullet W^r \bullet \gamma^r \, , \end{align}
(4)

where WrW^r is the screened interaction in channel rr, defined through the bosonic Dyson equation

Wr=F0+F0∘χ0r∘γ‾r∙Wr=F0+Wr∙γr∘χ0r∘F0 .\begin{align} W^r &= F_0 + F_0 \circ \chi^r_0 \circ \overline{\gamma}^r \bullet W^r \\ &= F_0 + W^r \bullet \gamma^r \circ \chi^r_0 \circ F_0 \, . \end{align}
(5)

Again, the screened interaction is UU-reducible in channel rr and, crucially, depends only on a single frequency or momentum (for an instantaneous or local bare interaction, respectively).

Defining the polarizations in channel rr as

Pr=γr∘χ0r∘1r=1r∘χ0r∘γ‾r ,\begin{align} P^r &= \gamma^r \circ \chi_0^r \circ \mathbf{1}^r \\ &= \mathbf{1}^r \circ \chi_0^r \circ \overline{\gamma}^r\, , \end{align}
(6)

the bosonic Dyson equation can be written more compactly as

Wr=F0+F0∙Pr∙Wr=F0+Wr∙Pr∙F0 .\begin{align} W^r = F_0 + F_0 \bullet P^r \bullet W^r = F_0 + W^r \bullet P^r \bullet F_0 \, . \end{align}
(7)

In summary, the equations to solve for the UU-reducible contributions of the vertex read,

Wr=F0+F0∙Pr∙Wr=F0+Wr∙Pr∙F0Pr=γr∘χ0r∘1r=1r∘χ0r∘γ‾rγr=1r+1r∘χ0r∘Trγ‾r=1r+Tr∘χ0r∘1rTr=F−γ‾r∙Wr∙γr .\begin{align} W^r &= F_0 + F_0 \bullet P^r \bullet W^r = F_0 + W^r \bullet P^r \bullet F_0 \\ P^r &= \gamma^r \circ \chi_0^r \circ \mathbf{1}^r = \mathbf{1}^r \circ \chi_0^r \circ \overline{\gamma}^r \\ \gamma^r &= \mathbf{1}^r + \mathbf{1}^r \circ \chi_0^r \circ T^r \\ \overline{\gamma}^r &= \mathbf{1}^r + T^r \circ \chi_0^r \circ \mathbf{1}^r \\ T^r &= F - \overline{\gamma}^r \bullet W^r \bullet \gamma^r\, . \end{align}
(8)

As in parquet theory, these equations can be solved self-consistently by iteration, starting from an initial guess. Again, this procedure requires knowledge of the UU-irreducible vertex ΛU\Lambda^{U} to compute TrT^r from FF, which is generally unknown. In practice, one therefore often resorts to approximations for ΛU\Lambda^{U}, such as the SBE approximation, discussed below.

Schwinger-Dyson equation in SBE form

Finally, the self-energy should again be included through the Schwinger-Dyson equation (SDE) as discussed in the chapter on parquet theory. In terms of SBE objects, it takes a simple form, since one can derive that

F0∘χ0r∘F=Wr∙γr−F0F∘χ0r∘F0=γ‾r∙Wr−F0 .\begin{align} F_0 \circ \chi_0^r \circ F &= W^r \bullet \gamma^r - F_0 \\ F \circ \chi_0^r \circ F_0 &= \overline{\gamma}^r \bullet W^r - F_0 \, . \end{align}
(9)
Explicit derivation

Using the decomposition of the UU-reducible diagrams into Hedin vertices and screened interactions, as well as the definition of the polarizations, and the Dyson equation for the screened interaction, we have

F0∘χ0r∘F=F0∘χ0r∘(Δr+Tr)=F0∘χ0r∘(γ‾r∙Wr∙γr+Tr)=F0∙1r∘χ0r∘γ‾r∙Wr∙γr+F0∙1r∘χ0r∘Tr=F0∙Pr∙Wr∙γr+F0∙γr−F0=Wr∙γr−F0∙γr+F0∙γr−F0=Wr∙γr−F0 ,\begin{align} F_0 \circ \chi_0^r \circ F &= F_0 \circ \chi_0^r \circ \left( \Delta^r + T^r \right) \\ &= F_0 \circ \chi_0^r \circ \left( \overline{\gamma}^r \bullet W^r \bullet \gamma^r + T^r \right) \\ &= F_0 \bullet \mathbf{1}^r \circ \chi_0^r \circ \overline{\gamma}^r \bullet W^r \bullet \gamma^r + F_0 \bullet \mathbf{1}^r \circ \chi_0^r \circ T^r \\ &= F_0 \bullet P^r \bullet W^r \bullet \gamma^r + F_0 \bullet \gamma^r - F_0 \\ &= W^r \bullet \gamma^r - F_0 \bullet \gamma^r + F_0 \bullet \gamma^r - F_0 \\ &= W^r \bullet \gamma^r - F_0 \, , \end{align}
(10)

and similarly,

F∘χ0r∘F0=(Δr+Tr)∘χ0r∘F0=(γ‾r∙Wr∙γr+Tr)∘χ0r∘F0=γ‾r∙Wr∙γr∘χ0r∘1r∙F0+Tr∘χ0r∘1r∙F0=γ‾r∙Wr∙Pr∙F0+γ‾r∙F0−F0=γ‾r∙Wr−γ‾r∙F0+γ‾r∙F0−F0=γ‾r∙Wr−F0 . ✓\begin{align} F \circ \chi_0^r \circ F_0 &= \left( \Delta^r + T^r \right) \circ \chi_0^r \circ F_0 \\ &= \left( \overline{\gamma}^r \bullet W^r \bullet \gamma^r + T^r \right) \circ \chi_0^r \circ F_0 \\ &= \overline{\gamma}^r \bullet W^r \bullet \gamma^r \circ \chi_0^r \circ \mathbf{1}^r \bullet F_0 + T^r \circ \chi_0^r \circ \mathbf{1}^r \bullet F_0 \\ &= \overline{\gamma}^r \bullet W^r \bullet P^r \bullet F_0 + \overline{\gamma}^r \bullet F_0 - F_0 \\ &= \overline{\gamma}^r \bullet W^r - \overline{\gamma}^r \bullet F_0 + \overline{\gamma}^r \bullet F_0 - F_0 \\ &= \overline{\gamma}^r \bullet W^r - F_0\, .\ \checkmark \end{align}
(11)

The SDE can therefore be written as

Σ=12ζ(F0+Wph‾∙γph‾)⋅G=ζ(Wpp∙γpp)⋅G=12G⋅(F0+γ‾ph∙Wph) .\begin{align} \Sigma &= \frac{1}{2} \zeta (F_0 + W^{\overline{ph}} \bullet \gamma^{\overline{ph}}) \cdot G \\ &= \zeta (W^{pp} \bullet \gamma^{pp}) \cdot G \\ &= \frac{1}{2} G \cdot (F_0 + \overline{\gamma}^{ph} \bullet W^{ph}) \, . \end{align}
(12)

SBE approximation

In the SBE approximation, one neglects the UU-irreducible vertex altogether ΛU≃0\Lambda^{U} \simeq 0. This approximation is motivated by the conjecture that, in many physical situations, the dominant contributions to the full vertex arise from the exchange of single bosonic fluctuations, while more complex processes involving multiple boson exchanges are less important. Since F≃∑rΔr−2F0F \simeq \sum_r \Delta^r - 2F_0 in this approximation, we have for the UU-irreducible vertex in channel rr,

Tr≃∑r′≠rΔr′−2F0=∑r′≠rγ‾r′∙Wr′∙γr′−2F0 .\begin{align} T^r &\simeq \sum_{r' \neq r} \Delta^{r'} - 2F_0 \\ &= \sum_{r' \neq r} \overline{\gamma}^{r'} \bullet W^{r'} \bullet \gamma^{r'} - 2F_0 \, . \end{align}
(13)

We note in passing, that the SBE formulation of the SDE is reminiscent of the equation for Σ\Sigma in the GWGW approximation. Indeed, the GWGW approximation can be seen as a further simplification of the SBE approximation, where the Hedin vertices are approximated by their lowest-order contribution, γr≃1r\gamma^r \simeq \mathbf{1}^r and γ‾r≃1r\overline{\gamma}^r \simeq \mathbf{1}^r. The self-energy is then given by

Σ≃ζWpp⋅G ,\begin{align} \Sigma \simeq \zeta W^{pp} \cdot G \, , \end{align}
(14)

which is precisely the GWGW expression for the self-energy.

Diagrammatic frameworks
Parquet theory
Advanced topics
Two-particle QFT in the real-frequency Keldysh formalism